math.sin on complex, imaginary part

Percentage Accurate: 53.6% → 99.7%

Time: 11.4s

Alternatives: 16

Speedup: 2.8×

• 49.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 53.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

C

double code(double re, double im) {
	return (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
end function

Java

public static double code(double re, double im) {
	return (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
}

Python

def code(re, im):
	return (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))

Julia

function code(re, im)
	return Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)))
end

MATLAB

function tmp = code(re, im)
	tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
end

Wolfram

code[re_, im_] := N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \cos re\\ t_1 := e^{-im} - e^{im}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 0.001\right):\\ \;\;\;\;t_0 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(im \cdot -2 + \left(-0.016666666666666666 \cdot {im}^{5} + -0.3333333333333333 \cdot {im}^{3}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (cos re))) (t_1 (- (exp (- im)) (exp im))))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 0.001)))
     (* t_0 t_1)
     (*
      t_0
      (+
       (* im -2.0)
       (+
        (* -0.016666666666666666 (pow im 5.0))
        (* -0.3333333333333333 (pow im 3.0))))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * cos(re);
	double t_1 = exp(-im) - exp(im);
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 0.001)) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_0 * ((im * -2.0) + ((-0.016666666666666666 * pow(im, 5.0)) + (-0.3333333333333333 * pow(im, 3.0))));
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * Math.cos(re);
	double t_1 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_1 <= -Double.POSITIVE_INFINITY) || !(t_1 <= 0.001)) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_0 * ((im * -2.0) + ((-0.016666666666666666 * Math.pow(im, 5.0)) + (-0.3333333333333333 * Math.pow(im, 3.0))));
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * math.cos(re)
	t_1 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_1 <= -math.inf) or not (t_1 <= 0.001):
		tmp = t_0 * t_1
	else:
		tmp = t_0 * ((im * -2.0) + ((-0.016666666666666666 * math.pow(im, 5.0)) + (-0.3333333333333333 * math.pow(im, 3.0))))
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * cos(re))
	t_1 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 0.001))
		tmp = Float64(t_0 * t_1);
	else
		tmp = Float64(t_0 * Float64(Float64(im * -2.0) + Float64(Float64(-0.016666666666666666 * (im ^ 5.0)) + Float64(-0.3333333333333333 * (im ^ 3.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * cos(re);
	t_1 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_1 <= -Inf) || ~((t_1 <= 0.001)))
		tmp = t_0 * t_1;
	else
		tmp = t_0 * ((im * -2.0) + ((-0.016666666666666666 * (im ^ 5.0)) + (-0.3333333333333333 * (im ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 0.001]], $MachinePrecision]], N[(t$95$0 * t$95$1), $MachinePrecision], N[(t$95$0 * N[(N[(im * -2.0), $MachinePrecision] + N[(N[(-0.016666666666666666 * N[Power[im, 5.0], $MachinePrecision]), $MachinePrecision] + N[(-0.3333333333333333 * N[Power[im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -inf.0 or 1e-3 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   if -inf.0 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 1e-3

   1. Initial program 6.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 6.3%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 6.3%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in im around 0 99.7%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.016666666666666666 \cdot {im}^{5} + -0.3333333333333333 \cdot {im}^{3}\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty \lor \neg \left(e^{-im} - e^{im} \leq 0.001\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(im \cdot -2 + \left(-0.016666666666666666 \cdot {im}^{5} + -0.3333333333333333 \cdot {im}^{3}\right)\right)\\ \end{array} \]

Alternative 2: 99.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-im} - e^{im}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 0.001\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (exp (- im)) (exp im))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 0.001)))
     (* (* 0.5 (cos re)) t_0)
     (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im)))))

C

double code(double re, double im) {
	double t_0 = exp(-im) - exp(im);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 0.001)) {
		tmp = (0.5 * cos(re)) * t_0;
	} else {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Java

public static double code(double re, double im) {
	double t_0 = Math.exp(-im) - Math.exp(im);
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 0.001)) {
		tmp = (0.5 * Math.cos(re)) * t_0;
	} else {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = math.exp(-im) - math.exp(im)
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 0.001):
		tmp = (0.5 * math.cos(re)) * t_0
	else:
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(exp(Float64(-im)) - exp(im))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 0.001))
		tmp = Float64(Float64(0.5 * cos(re)) * t_0);
	else
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = exp(-im) - exp(im);
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 0.001)))
		tmp = (0.5 * cos(re)) * t_0;
	else
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 0.001]], $MachinePrecision]], N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < -inf.0 or 1e-3 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im))

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   if -inf.0 < (-.f64 (exp.f64 (-.f64 0 im)) (exp.f64 im)) < 1e-3

   1. Initial program 6.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 6.3%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 6.3%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in im around 0 99.7%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 99.7%

         \[\leadsto -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \color{blue}{\left(-\cos re \cdot im\right)} \]

      2. unsub-neg 99.7%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) - \cos re \cdot im} \]

      3. *-commutative 99.7%

         \[\leadsto \color{blue}{\left(\cos re \cdot {im}^{3}\right) \cdot -0.16666666666666666} - \cos re \cdot im \]

      4. associate-*l* 99.7%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)} - \cos re \cdot im \]

      5. distribute-lft-out-- 99.7%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   6. Simplified 99.7%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{-im} - e^{im} \leq -\infty \lor \neg \left(e^{-im} - e^{im} \leq 0.001\right):\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \end{array} \]

Alternative 3: 87.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := {im}^{3} \cdot -0.16666666666666666 - im\\ t_2 := t_1 \cdot \left(\left(re \cdot re\right) \cdot -0.5 + 1\right)\\ \mathbf{if}\;im \leq -3 \cdot 10^{+205}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;im \leq -0.00132:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.00044:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+189}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 10^{+219}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (- (exp (- im)) (exp im))))
        (t_1 (- (* (pow im 3.0) -0.16666666666666666) im))
        (t_2 (* t_1 (+ (* (* re re) -0.5) 1.0))))
   (if (<= im -3e+205)
     t_2
     (if (<= im -0.00132)
       t_0
       (if (<= im 0.00044)
         (* im (- (cos re)))
         (if (<= im 1.05e+189) t_0 (if (<= im 1e+219) t_2 t_1)))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(-im) - exp(im));
	double t_1 = (pow(im, 3.0) * -0.16666666666666666) - im;
	double t_2 = t_1 * (((re * re) * -0.5) + 1.0);
	double tmp;
	if (im <= -3e+205) {
		tmp = t_2;
	} else if (im <= -0.00132) {
		tmp = t_0;
	} else if (im <= 0.00044) {
		tmp = im * -cos(re);
	} else if (im <= 1.05e+189) {
		tmp = t_0;
	} else if (im <= 1e+219) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 0.5d0 * (exp(-im) - exp(im))
    t_1 = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    t_2 = t_1 * (((re * re) * (-0.5d0)) + 1.0d0)
    if (im <= (-3d+205)) then
        tmp = t_2
    else if (im <= (-0.00132d0)) then
        tmp = t_0
    else if (im <= 0.00044d0) then
        tmp = im * -cos(re)
    else if (im <= 1.05d+189) then
        tmp = t_0
    else if (im <= 1d+219) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (Math.exp(-im) - Math.exp(im));
	double t_1 = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	double t_2 = t_1 * (((re * re) * -0.5) + 1.0);
	double tmp;
	if (im <= -3e+205) {
		tmp = t_2;
	} else if (im <= -0.00132) {
		tmp = t_0;
	} else if (im <= 0.00044) {
		tmp = im * -Math.cos(re);
	} else if (im <= 1.05e+189) {
		tmp = t_0;
	} else if (im <= 1e+219) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.exp(-im) - math.exp(im))
	t_1 = (math.pow(im, 3.0) * -0.16666666666666666) - im
	t_2 = t_1 * (((re * re) * -0.5) + 1.0)
	tmp = 0
	if im <= -3e+205:
		tmp = t_2
	elif im <= -0.00132:
		tmp = t_0
	elif im <= 0.00044:
		tmp = im * -math.cos(re)
	elif im <= 1.05e+189:
		tmp = t_0
	elif im <= 1e+219:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)))
	t_1 = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im)
	t_2 = Float64(t_1 * Float64(Float64(Float64(re * re) * -0.5) + 1.0))
	tmp = 0.0
	if (im <= -3e+205)
		tmp = t_2;
	elseif (im <= -0.00132)
		tmp = t_0;
	elseif (im <= 0.00044)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 1.05e+189)
		tmp = t_0;
	elseif (im <= 1e+219)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (exp(-im) - exp(im));
	t_1 = ((im ^ 3.0) * -0.16666666666666666) - im;
	t_2 = t_1 * (((re * re) * -0.5) + 1.0);
	tmp = 0.0;
	if (im <= -3e+205)
		tmp = t_2;
	elseif (im <= -0.00132)
		tmp = t_0;
	elseif (im <= 0.00044)
		tmp = im * -cos(re);
	elseif (im <= 1.05e+189)
		tmp = t_0;
	elseif (im <= 1e+219)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -3e+205], t$95$2, If[LessEqual[im, -0.00132], t$95$0, If[LessEqual[im, 0.00044], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 1.05e+189], t$95$0, If[LessEqual[im, 1e+219], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if im < -2.9999999999999999e205 or 1.04999999999999996e189 < im < 9.99999999999999965e218

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \color{blue}{\left(-\cos re \cdot im\right)} \]

      2. unsub-neg 100.0%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) - \cos re \cdot im} \]

      3. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\cos re \cdot {im}^{3}\right) \cdot -0.16666666666666666} - \cos re \cdot im \]

      4. associate-*l* 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)} - \cos re \cdot im \]

      5. distribute-lft-out-- 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   7. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + -0.16666666666666666 \cdot {im}^{3}\right) - im} \]
   8. Step-by-step derivation

      1. associate--l+ 0.0%

         \[\leadsto \color{blue}{-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      2. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.5 \cdot {re}^{2}\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} + \left(-0.16666666666666666 \cdot {im}^{3} - im\right) \]

      3. distribute-lft1-in 93.1%

         \[\leadsto \color{blue}{\left(-0.5 \cdot {re}^{2} + 1\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      4. *-commutative 93.1%

         \[\leadsto \left(\color{blue}{{re}^{2} \cdot -0.5} + 1\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right) \]

      5. unpow2 93.1%

         \[\leadsto \left(\color{blue}{\left(re \cdot re\right)} \cdot -0.5 + 1\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right) \]

      6. *-commutative 93.1%

         \[\leadsto \left(\left(re \cdot re\right) \cdot -0.5 + 1\right) \cdot \left(\color{blue}{{im}^{3} \cdot -0.16666666666666666} - im\right) \]

   9. Simplified 93.1%

      \[\leadsto \color{blue}{\left(\left(re \cdot re\right) \cdot -0.5 + 1\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   if -2.9999999999999999e205 < im < -0.00132 or 4.40000000000000016e-4 < im < 1.04999999999999996e189

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in re around 0 83.9%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-im} - e^{im}\right)} \]

   if -0.00132 < im < 4.40000000000000016e-4

   1. Initial program 6.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 6.3%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 6.3%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in im around 0 99.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 99.4%

         \[\leadsto \color{blue}{-\cos re \cdot im} \]

      2. *-commutative 99.4%

         \[\leadsto -\color{blue}{im \cdot \cos re} \]

      3. distribute-lft-neg-in 99.4%

         \[\leadsto \color{blue}{\left(-im\right) \cdot \cos re} \]

   6. Simplified 99.4%

      \[\leadsto \color{blue}{\left(-im\right) \cdot \cos re} \]

   if 9.99999999999999965e218 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \color{blue}{\left(-\cos re \cdot im\right)} \]

      2. unsub-neg 100.0%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) - \cos re \cdot im} \]

      3. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\cos re \cdot {im}^{3}\right) \cdot -0.16666666666666666} - \cos re \cdot im \]

      4. associate-*l* 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)} - \cos re \cdot im \]

      5. distribute-lft-out-- 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   7. Taylor expanded in re around 0 95.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]
   8. Step-by-step derivation

      1. *-commutative 95.0%

         \[\leadsto \color{blue}{{im}^{3} \cdot -0.16666666666666666} - im \]

   9. Simplified 95.0%

      \[\leadsto \color{blue}{{im}^{3} \cdot -0.16666666666666666 - im} \]
3. Recombined 4 regimes into one program.

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -3 \cdot 10^{+205}:\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.5 + 1\right)\\ \mathbf{elif}\;im \leq -0.00132:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 0.00044:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 1.05 \cdot 10^{+189}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 10^{+219}:\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.5 + 1\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \end{array} \]

Alternative 4: 96.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := {im}^{5} \cdot \left(\cos re \cdot -0.008333333333333333\right)\\ \mathbf{if}\;im \leq -6.5 \cdot 10^{+97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.00132:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.029:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (- (exp (- im)) (exp im))))
        (t_1 (* (pow im 5.0) (* (cos re) -0.008333333333333333))))
   (if (<= im -6.5e+97)
     t_1
     (if (<= im -0.00132)
       t_0
       (if (<= im 0.029) (* im (- (cos re))) (if (<= im 5e+61) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(-im) - exp(im));
	double t_1 = pow(im, 5.0) * (cos(re) * -0.008333333333333333);
	double tmp;
	if (im <= -6.5e+97) {
		tmp = t_1;
	} else if (im <= -0.00132) {
		tmp = t_0;
	} else if (im <= 0.029) {
		tmp = im * -cos(re);
	} else if (im <= 5e+61) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (exp(-im) - exp(im))
    t_1 = (im ** 5.0d0) * (cos(re) * (-0.008333333333333333d0))
    if (im <= (-6.5d+97)) then
        tmp = t_1
    else if (im <= (-0.00132d0)) then
        tmp = t_0
    else if (im <= 0.029d0) then
        tmp = im * -cos(re)
    else if (im <= 5d+61) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (Math.exp(-im) - Math.exp(im));
	double t_1 = Math.pow(im, 5.0) * (Math.cos(re) * -0.008333333333333333);
	double tmp;
	if (im <= -6.5e+97) {
		tmp = t_1;
	} else if (im <= -0.00132) {
		tmp = t_0;
	} else if (im <= 0.029) {
		tmp = im * -Math.cos(re);
	} else if (im <= 5e+61) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.exp(-im) - math.exp(im))
	t_1 = math.pow(im, 5.0) * (math.cos(re) * -0.008333333333333333)
	tmp = 0
	if im <= -6.5e+97:
		tmp = t_1
	elif im <= -0.00132:
		tmp = t_0
	elif im <= 0.029:
		tmp = im * -math.cos(re)
	elif im <= 5e+61:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)))
	t_1 = Float64((im ^ 5.0) * Float64(cos(re) * -0.008333333333333333))
	tmp = 0.0
	if (im <= -6.5e+97)
		tmp = t_1;
	elseif (im <= -0.00132)
		tmp = t_0;
	elseif (im <= 0.029)
		tmp = Float64(im * Float64(-cos(re)));
	elseif (im <= 5e+61)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (exp(-im) - exp(im));
	t_1 = (im ^ 5.0) * (cos(re) * -0.008333333333333333);
	tmp = 0.0;
	if (im <= -6.5e+97)
		tmp = t_1;
	elseif (im <= -0.00132)
		tmp = t_0;
	elseif (im <= 0.029)
		tmp = im * -cos(re);
	elseif (im <= 5e+61)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[im, 5.0], $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -6.5e+97], t$95$1, If[LessEqual[im, -0.00132], t$95$0, If[LessEqual[im, 0.029], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], If[LessEqual[im, 5e+61], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -6.4999999999999999e97 or 5.00000000000000018e61 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.016666666666666666 \cdot {im}^{5} + -0.3333333333333333 \cdot {im}^{3}\right)\right)} \]

   5. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\cos re \cdot {im}^{5}\right) \cdot -0.008333333333333333} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{\left({im}^{5} \cdot \cos re\right)} \cdot -0.008333333333333333 \]

      3. associate-*l* 100.0%

         \[\leadsto \color{blue}{{im}^{5} \cdot \left(\cos re \cdot -0.008333333333333333\right)} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{{im}^{5} \cdot \left(\cos re \cdot -0.008333333333333333\right)} \]

   if -6.4999999999999999e97 < im < -0.00132 or 0.0290000000000000015 < im < 5.00000000000000018e61

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in re around 0 90.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-im} - e^{im}\right)} \]

   if -0.00132 < im < 0.0290000000000000015

   1. Initial program 6.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 6.3%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 6.3%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in im around 0 99.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 99.4%

         \[\leadsto \color{blue}{-\cos re \cdot im} \]

      2. *-commutative 99.4%

         \[\leadsto -\color{blue}{im \cdot \cos re} \]

      3. distribute-lft-neg-in 99.4%

         \[\leadsto \color{blue}{\left(-im\right) \cdot \cos re} \]

   6. Simplified 99.4%

      \[\leadsto \color{blue}{\left(-im\right) \cdot \cos re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6.5 \cdot 10^{+97}:\\ \;\;\;\;{im}^{5} \cdot \left(\cos re \cdot -0.008333333333333333\right)\\ \mathbf{elif}\;im \leq -0.00132:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 0.029:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{elif}\;im \leq 5 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{5} \cdot \left(\cos re \cdot -0.008333333333333333\right)\\ \end{array} \]

Alternative 5: 96.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 \cdot \left(e^{-im} - e^{im}\right)\\ t_1 := {im}^{5} \cdot \left(\cos re \cdot -0.008333333333333333\right)\\ \mathbf{if}\;im \leq -6.5 \cdot 10^{+97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;im \leq -0.068:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq 0.052:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* 0.5 (- (exp (- im)) (exp im))))
        (t_1 (* (pow im 5.0) (* (cos re) -0.008333333333333333))))
   (if (<= im -6.5e+97)
     t_1
     (if (<= im -0.068)
       t_0
       (if (<= im 0.052)
         (* (cos re) (- (* (pow im 3.0) -0.16666666666666666) im))
         (if (<= im 4.5e+61) t_0 t_1))))))

C

double code(double re, double im) {
	double t_0 = 0.5 * (exp(-im) - exp(im));
	double t_1 = pow(im, 5.0) * (cos(re) * -0.008333333333333333);
	double tmp;
	if (im <= -6.5e+97) {
		tmp = t_1;
	} else if (im <= -0.068) {
		tmp = t_0;
	} else if (im <= 0.052) {
		tmp = cos(re) * ((pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 4.5e+61) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 0.5d0 * (exp(-im) - exp(im))
    t_1 = (im ** 5.0d0) * (cos(re) * (-0.008333333333333333d0))
    if (im <= (-6.5d+97)) then
        tmp = t_1
    else if (im <= (-0.068d0)) then
        tmp = t_0
    else if (im <= 0.052d0) then
        tmp = cos(re) * (((im ** 3.0d0) * (-0.16666666666666666d0)) - im)
    else if (im <= 4.5d+61) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = 0.5 * (Math.exp(-im) - Math.exp(im));
	double t_1 = Math.pow(im, 5.0) * (Math.cos(re) * -0.008333333333333333);
	double tmp;
	if (im <= -6.5e+97) {
		tmp = t_1;
	} else if (im <= -0.068) {
		tmp = t_0;
	} else if (im <= 0.052) {
		tmp = Math.cos(re) * ((Math.pow(im, 3.0) * -0.16666666666666666) - im);
	} else if (im <= 4.5e+61) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = 0.5 * (math.exp(-im) - math.exp(im))
	t_1 = math.pow(im, 5.0) * (math.cos(re) * -0.008333333333333333)
	tmp = 0
	if im <= -6.5e+97:
		tmp = t_1
	elif im <= -0.068:
		tmp = t_0
	elif im <= 0.052:
		tmp = math.cos(re) * ((math.pow(im, 3.0) * -0.16666666666666666) - im)
	elif im <= 4.5e+61:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(re, im)
	t_0 = Float64(0.5 * Float64(exp(Float64(-im)) - exp(im)))
	t_1 = Float64((im ^ 5.0) * Float64(cos(re) * -0.008333333333333333))
	tmp = 0.0
	if (im <= -6.5e+97)
		tmp = t_1;
	elseif (im <= -0.068)
		tmp = t_0;
	elseif (im <= 0.052)
		tmp = Float64(cos(re) * Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im));
	elseif (im <= 4.5e+61)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = 0.5 * (exp(-im) - exp(im));
	t_1 = (im ^ 5.0) * (cos(re) * -0.008333333333333333);
	tmp = 0.0;
	if (im <= -6.5e+97)
		tmp = t_1;
	elseif (im <= -0.068)
		tmp = t_0;
	elseif (im <= 0.052)
		tmp = cos(re) * (((im ^ 3.0) * -0.16666666666666666) - im);
	elseif (im <= 4.5e+61)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(0.5 * N[(N[Exp[(-im)], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[im, 5.0], $MachinePrecision] * N[(N[Cos[re], $MachinePrecision] * -0.008333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[im, -6.5e+97], t$95$1, If[LessEqual[im, -0.068], t$95$0, If[LessEqual[im, 0.052], N[(N[Cos[re], $MachinePrecision] * N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, 4.5e+61], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if im < -6.4999999999999999e97 or 4.5e61 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \color{blue}{\left(-2 \cdot im + \left(-0.016666666666666666 \cdot {im}^{5} + -0.3333333333333333 \cdot {im}^{3}\right)\right)} \]

   5. Taylor expanded in im around inf 100.0%

      \[\leadsto \color{blue}{-0.008333333333333333 \cdot \left(\cos re \cdot {im}^{5}\right)} \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\cos re \cdot {im}^{5}\right) \cdot -0.008333333333333333} \]

      2. *-commutative 100.0%

         \[\leadsto \color{blue}{\left({im}^{5} \cdot \cos re\right)} \cdot -0.008333333333333333 \]

      3. associate-*l* 100.0%

         \[\leadsto \color{blue}{{im}^{5} \cdot \left(\cos re \cdot -0.008333333333333333\right)} \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{{im}^{5} \cdot \left(\cos re \cdot -0.008333333333333333\right)} \]

   if -6.4999999999999999e97 < im < -0.068000000000000005 or 0.0519999999999999976 < im < 4.5e61

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in re around 0 90.3%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-im} - e^{im}\right)} \]

   if -0.068000000000000005 < im < 0.0519999999999999976

   1. Initial program 6.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 6.3%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 6.3%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in im around 0 99.7%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 99.7%

         \[\leadsto -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \color{blue}{\left(-\cos re \cdot im\right)} \]

      2. unsub-neg 99.7%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) - \cos re \cdot im} \]

      3. *-commutative 99.7%

         \[\leadsto \color{blue}{\left(\cos re \cdot {im}^{3}\right) \cdot -0.16666666666666666} - \cos re \cdot im \]

      4. associate-*l* 99.7%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)} - \cos re \cdot im \]

      5. distribute-lft-out-- 99.7%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   6. Simplified 99.7%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -6.5 \cdot 10^{+97}:\\ \;\;\;\;{im}^{5} \cdot \left(\cos re \cdot -0.008333333333333333\right)\\ \mathbf{elif}\;im \leq -0.068:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{elif}\;im \leq 0.052:\\ \;\;\;\;\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)\\ \mathbf{elif}\;im \leq 4.5 \cdot 10^{+61}:\\ \;\;\;\;0.5 \cdot \left(e^{-im} - e^{im}\right)\\ \mathbf{else}:\\ \;\;\;\;{im}^{5} \cdot \left(\cos re \cdot -0.008333333333333333\right)\\ \end{array} \]

Alternative 6: 75.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{if}\;im \leq -5 \cdot 10^{+214}:\\ \;\;\;\;t_0 \cdot \left(\left(re \cdot re\right) \cdot -0.5 + 1\right)\\ \mathbf{elif}\;im \leq -2.65 \cdot 10^{+44} \lor \neg \left(im \leq 8.5 \cdot 10^{+48}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (- (* (pow im 3.0) -0.16666666666666666) im)))
   (if (<= im -5e+214)
     (* t_0 (+ (* (* re re) -0.5) 1.0))
     (if (or (<= im -2.65e+44) (not (<= im 8.5e+48)))
       t_0
       (* im (- (cos re)))))))

C

double code(double re, double im) {
	double t_0 = (pow(im, 3.0) * -0.16666666666666666) - im;
	double tmp;
	if (im <= -5e+214) {
		tmp = t_0 * (((re * re) * -0.5) + 1.0);
	} else if ((im <= -2.65e+44) || !(im <= 8.5e+48)) {
		tmp = t_0;
	} else {
		tmp = im * -cos(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    if (im <= (-5d+214)) then
        tmp = t_0 * (((re * re) * (-0.5d0)) + 1.0d0)
    else if ((im <= (-2.65d+44)) .or. (.not. (im <= 8.5d+48))) then
        tmp = t_0
    else
        tmp = im * -cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	double tmp;
	if (im <= -5e+214) {
		tmp = t_0 * (((re * re) * -0.5) + 1.0);
	} else if ((im <= -2.65e+44) || !(im <= 8.5e+48)) {
		tmp = t_0;
	} else {
		tmp = im * -Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (math.pow(im, 3.0) * -0.16666666666666666) - im
	tmp = 0
	if im <= -5e+214:
		tmp = t_0 * (((re * re) * -0.5) + 1.0)
	elif (im <= -2.65e+44) or not (im <= 8.5e+48):
		tmp = t_0
	else:
		tmp = im * -math.cos(re)
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im)
	tmp = 0.0
	if (im <= -5e+214)
		tmp = Float64(t_0 * Float64(Float64(Float64(re * re) * -0.5) + 1.0));
	elseif ((im <= -2.65e+44) || !(im <= 8.5e+48))
		tmp = t_0;
	else
		tmp = Float64(im * Float64(-cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = ((im ^ 3.0) * -0.16666666666666666) - im;
	tmp = 0.0;
	if (im <= -5e+214)
		tmp = t_0 * (((re * re) * -0.5) + 1.0);
	elseif ((im <= -2.65e+44) || ~((im <= 8.5e+48)))
		tmp = t_0;
	else
		tmp = im * -cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision]}, If[LessEqual[im, -5e+214], N[(t$95$0 * N[(N[(N[(re * re), $MachinePrecision] * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[im, -2.65e+44], N[Not[LessEqual[im, 8.5e+48]], $MachinePrecision]], t$95$0, N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if im < -4.99999999999999953e214

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in im around 0 100.0%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 100.0%

         \[\leadsto -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \color{blue}{\left(-\cos re \cdot im\right)} \]

      2. unsub-neg 100.0%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) - \cos re \cdot im} \]

      3. *-commutative 100.0%

         \[\leadsto \color{blue}{\left(\cos re \cdot {im}^{3}\right) \cdot -0.16666666666666666} - \cos re \cdot im \]

      4. associate-*l* 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)} - \cos re \cdot im \]

      5. distribute-lft-out-- 100.0%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   7. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + -0.16666666666666666 \cdot {im}^{3}\right) - im} \]
   8. Step-by-step derivation

      1. associate--l+ 0.0%

         \[\leadsto \color{blue}{-0.5 \cdot \left({re}^{2} \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)\right) + \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      2. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.5 \cdot {re}^{2}\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} + \left(-0.16666666666666666 \cdot {im}^{3} - im\right) \]

      3. distribute-lft1-in 90.5%

         \[\leadsto \color{blue}{\left(-0.5 \cdot {re}^{2} + 1\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right)} \]

      4. *-commutative 90.5%

         \[\leadsto \left(\color{blue}{{re}^{2} \cdot -0.5} + 1\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right) \]

      5. unpow2 90.5%

         \[\leadsto \left(\color{blue}{\left(re \cdot re\right)} \cdot -0.5 + 1\right) \cdot \left(-0.16666666666666666 \cdot {im}^{3} - im\right) \]

      6. *-commutative 90.5%

         \[\leadsto \left(\left(re \cdot re\right) \cdot -0.5 + 1\right) \cdot \left(\color{blue}{{im}^{3} \cdot -0.16666666666666666} - im\right) \]

   9. Simplified 90.5%

      \[\leadsto \color{blue}{\left(\left(re \cdot re\right) \cdot -0.5 + 1\right) \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   if -4.99999999999999953e214 < im < -2.65e44 or 8.5000000000000001e48 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in im around 0 74.4%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 74.4%

         \[\leadsto -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \color{blue}{\left(-\cos re \cdot im\right)} \]

      2. unsub-neg 74.4%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) - \cos re \cdot im} \]

      3. *-commutative 74.4%

         \[\leadsto \color{blue}{\left(\cos re \cdot {im}^{3}\right) \cdot -0.16666666666666666} - \cos re \cdot im \]

      4. associate-*l* 74.4%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)} - \cos re \cdot im \]

      5. distribute-lft-out-- 74.4%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   6. Simplified 74.4%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   7. Taylor expanded in re around 0 60.4%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]
   8. Step-by-step derivation

      1. *-commutative 60.4%

         \[\leadsto \color{blue}{{im}^{3} \cdot -0.16666666666666666} - im \]

   9. Simplified 60.4%

      \[\leadsto \color{blue}{{im}^{3} \cdot -0.16666666666666666 - im} \]

   if -2.65e44 < im < 8.5000000000000001e48

   1. Initial program 15.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 15.3%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 15.3%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in im around 0 90.2%

      \[\leadsto \color{blue}{-1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 90.2%

         \[\leadsto \color{blue}{-\cos re \cdot im} \]

      2. *-commutative 90.2%

         \[\leadsto -\color{blue}{im \cdot \cos re} \]

      3. distribute-lft-neg-in 90.2%

         \[\leadsto \color{blue}{\left(-im\right) \cdot \cos re} \]

   6. Simplified 90.2%

      \[\leadsto \color{blue}{\left(-im\right) \cdot \cos re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5 \cdot 10^{+214}:\\ \;\;\;\;\left({im}^{3} \cdot -0.16666666666666666 - im\right) \cdot \left(\left(re \cdot re\right) \cdot -0.5 + 1\right)\\ \mathbf{elif}\;im \leq -2.65 \cdot 10^{+44} \lor \neg \left(im \leq 8.5 \cdot 10^{+48}\right):\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 7: 57.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -7.6 \cdot 10^{+167}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot re\right)\right) - im\\ \mathbf{elif}\;im \leq -4.1 \cdot 10^{+18}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \mathbf{elif}\;im \leq 680:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -7.6e+167)
   (- (* 0.5 (* re (* im re))) im)
   (if (<= im -4.1e+18)
     (* (* re re) 0.75)
     (if (<= im 680.0) (* im (- (cos re))) (* (* re re) -6.75)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -7.6e+167) {
		tmp = (0.5 * (re * (im * re))) - im;
	} else if (im <= -4.1e+18) {
		tmp = (re * re) * 0.75;
	} else if (im <= 680.0) {
		tmp = im * -cos(re);
	} else {
		tmp = (re * re) * -6.75;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-7.6d+167)) then
        tmp = (0.5d0 * (re * (im * re))) - im
    else if (im <= (-4.1d+18)) then
        tmp = (re * re) * 0.75d0
    else if (im <= 680.0d0) then
        tmp = im * -cos(re)
    else
        tmp = (re * re) * (-6.75d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -7.6e+167) {
		tmp = (0.5 * (re * (im * re))) - im;
	} else if (im <= -4.1e+18) {
		tmp = (re * re) * 0.75;
	} else if (im <= 680.0) {
		tmp = im * -Math.cos(re);
	} else {
		tmp = (re * re) * -6.75;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -7.6e+167:
		tmp = (0.5 * (re * (im * re))) - im
	elif im <= -4.1e+18:
		tmp = (re * re) * 0.75
	elif im <= 680.0:
		tmp = im * -math.cos(re)
	else:
		tmp = (re * re) * -6.75
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -7.6e+167)
		tmp = Float64(Float64(0.5 * Float64(re * Float64(im * re))) - im);
	elseif (im <= -4.1e+18)
		tmp = Float64(Float64(re * re) * 0.75);
	elseif (im <= 680.0)
		tmp = Float64(im * Float64(-cos(re)));
	else
		tmp = Float64(Float64(re * re) * -6.75);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -7.6e+167)
		tmp = (0.5 * (re * (im * re))) - im;
	elseif (im <= -4.1e+18)
		tmp = (re * re) * 0.75;
	elseif (im <= 680.0)
		tmp = im * -cos(re);
	else
		tmp = (re * re) * -6.75;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -7.6e+167], N[(N[(0.5 * N[(re * N[(im * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[im, -4.1e+18], N[(N[(re * re), $MachinePrecision] * 0.75), $MachinePrecision], If[LessEqual[im, 680.0], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision], N[(N[(re * re), $MachinePrecision] * -6.75), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < -7.59999999999999987e167

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in im around 0 7.0%

      \[\leadsto \color{blue}{-1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 7.0%

         \[\leadsto \color{blue}{-\cos re \cdot im} \]

      2. *-commutative 7.0%

         \[\leadsto -\color{blue}{im \cdot \cos re} \]

      3. distribute-lft-neg-in 7.0%

         \[\leadsto \color{blue}{\left(-im\right) \cdot \cos re} \]

   6. Simplified 7.0%

      \[\leadsto \color{blue}{\left(-im\right) \cdot \cos re} \]

   7. Taylor expanded in re around 0 43.1%

      \[\leadsto \color{blue}{-1 \cdot im + 0.5 \cdot \left({re}^{2} \cdot im\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 43.1%

         \[\leadsto \color{blue}{\left(-im\right)} + 0.5 \cdot \left({re}^{2} \cdot im\right) \]

      2. +-commutative 43.1%

         \[\leadsto \color{blue}{0.5 \cdot \left({re}^{2} \cdot im\right) + \left(-im\right)} \]

      3. unsub-neg 43.1%

         \[\leadsto \color{blue}{0.5 \cdot \left({re}^{2} \cdot im\right) - im} \]

      4. unpow2 43.1%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot im\right) - im \]

      5. associate-*l* 43.1%

         \[\leadsto 0.5 \cdot \color{blue}{\left(re \cdot \left(re \cdot im\right)\right)} - im \]

   9. Simplified 43.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot \left(re \cdot im\right)\right) - im} \]

   if -7.59999999999999987e167 < im < -4.1e18

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right) + 0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.25 \cdot \color{blue}{\left({re}^{2} \cdot \left(e^{-im} - e^{im}\right)\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      2. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{-im} - e^{im}\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      3. distribute-rgt-out 55.6%

         \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      4. +-commutative 55.6%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. *-commutative 55.6%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{{re}^{2} \cdot -0.25}\right) \]

      6. unpow2 55.6%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{\left(re \cdot re\right)} \cdot -0.25\right) \]

      7. associate-*l* 55.6%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{re \cdot \left(re \cdot -0.25\right)}\right) \]

   6. Simplified 55.6%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)} \]

   8. Applied egg-rr 26.3%

      \[\leadsto \color{blue}{-3} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

   9. Taylor expanded in re around inf 26.9%

      \[\leadsto \color{blue}{0.75 \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. *-commutative 26.9%

          \[\leadsto \color{blue}{{re}^{2} \cdot 0.75} \]

       2. unpow2 26.9%

          \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot 0.75 \]

   11. Simplified 26.9%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot 0.75} \]

   if -4.1e18 < im < 680

   1. Initial program 8.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 8.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 8.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in im around 0 97.7%

      \[\leadsto \color{blue}{-1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 97.7%

         \[\leadsto \color{blue}{-\cos re \cdot im} \]

      2. *-commutative 97.7%

         \[\leadsto -\color{blue}{im \cdot \cos re} \]

      3. distribute-lft-neg-in 97.7%

         \[\leadsto \color{blue}{\left(-im\right) \cdot \cos re} \]

   6. Simplified 97.7%

      \[\leadsto \color{blue}{\left(-im\right) \cdot \cos re} \]

   if 680 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right) + 0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.25 \cdot \color{blue}{\left({re}^{2} \cdot \left(e^{-im} - e^{im}\right)\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      2. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{-im} - e^{im}\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      3. distribute-rgt-out 71.6%

         \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      4. +-commutative 71.6%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. *-commutative 71.6%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{{re}^{2} \cdot -0.25}\right) \]

      6. unpow2 71.6%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{\left(re \cdot re\right)} \cdot -0.25\right) \]

      7. associate-*l* 71.6%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{re \cdot \left(re \cdot -0.25\right)}\right) \]

   6. Simplified 71.6%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)} \]

   7. Taylor expanded in re around inf 16.2%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 16.2%

         \[\leadsto \color{blue}{\left(-0.25 \cdot \left(e^{-im} - e^{im}\right)\right) \cdot {re}^{2}} \]

      2. *-commutative 16.2%

         \[\leadsto \color{blue}{{re}^{2} \cdot \left(-0.25 \cdot \left(e^{-im} - e^{im}\right)\right)} \]

      3. unpow2 16.2%

         \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot \left(-0.25 \cdot \left(e^{-im} - e^{im}\right)\right) \]

   9. Simplified 16.2%

      \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(-0.25 \cdot \left(e^{-im} - e^{im}\right)\right)} \]

   11. Applied egg-rr 19.2%

       \[\leadsto \left(re \cdot re\right) \cdot \left(-0.25 \cdot \color{blue}{27}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -7.6 \cdot 10^{+167}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot re\right)\right) - im\\ \mathbf{elif}\;im \leq -4.1 \cdot 10^{+18}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \mathbf{elif}\;im \leq 680:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \end{array} \]

Alternative 8: 75.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.32 \cdot 10^{+47} \lor \neg \left(im \leq 9 \cdot 10^{+46}\right):\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= im -1.32e+47) (not (<= im 9e+46)))
   (- (* (pow im 3.0) -0.16666666666666666) im)
   (* im (- (cos re)))))

C

double code(double re, double im) {
	double tmp;
	if ((im <= -1.32e+47) || !(im <= 9e+46)) {
		tmp = (pow(im, 3.0) * -0.16666666666666666) - im;
	} else {
		tmp = im * -cos(re);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((im <= (-1.32d+47)) .or. (.not. (im <= 9d+46))) then
        tmp = ((im ** 3.0d0) * (-0.16666666666666666d0)) - im
    else
        tmp = im * -cos(re)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((im <= -1.32e+47) || !(im <= 9e+46)) {
		tmp = (Math.pow(im, 3.0) * -0.16666666666666666) - im;
	} else {
		tmp = im * -Math.cos(re);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (im <= -1.32e+47) or not (im <= 9e+46):
		tmp = (math.pow(im, 3.0) * -0.16666666666666666) - im
	else:
		tmp = im * -math.cos(re)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((im <= -1.32e+47) || !(im <= 9e+46))
		tmp = Float64(Float64((im ^ 3.0) * -0.16666666666666666) - im);
	else
		tmp = Float64(im * Float64(-cos(re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((im <= -1.32e+47) || ~((im <= 9e+46)))
		tmp = ((im ^ 3.0) * -0.16666666666666666) - im;
	else
		tmp = im * -cos(re);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[im, -1.32e+47], N[Not[LessEqual[im, 9e+46]], $MachinePrecision]], N[(N[(N[Power[im, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision] - im), $MachinePrecision], N[(im * (-N[Cos[re], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if im < -1.31999999999999992e47 or 9.00000000000000019e46 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in im around 0 78.5%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + -1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 78.5%

         \[\leadsto -0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) + \color{blue}{\left(-\cos re \cdot im\right)} \]

      2. unsub-neg 78.5%

         \[\leadsto \color{blue}{-0.16666666666666666 \cdot \left(\cos re \cdot {im}^{3}\right) - \cos re \cdot im} \]

      3. *-commutative 78.5%

         \[\leadsto \color{blue}{\left(\cos re \cdot {im}^{3}\right) \cdot -0.16666666666666666} - \cos re \cdot im \]

      4. associate-*l* 78.5%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666\right)} - \cos re \cdot im \]

      5. distribute-lft-out-- 78.5%

         \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   6. Simplified 78.5%

      \[\leadsto \color{blue}{\cos re \cdot \left({im}^{3} \cdot -0.16666666666666666 - im\right)} \]

   7. Taylor expanded in re around 0 59.8%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {im}^{3} - im} \]
   8. Step-by-step derivation

      1. *-commutative 59.8%

         \[\leadsto \color{blue}{{im}^{3} \cdot -0.16666666666666666} - im \]

   9. Simplified 59.8%

      \[\leadsto \color{blue}{{im}^{3} \cdot -0.16666666666666666 - im} \]

   if -1.31999999999999992e47 < im < 9.00000000000000019e46

   1. Initial program 15.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 15.3%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 15.3%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in im around 0 90.2%

      \[\leadsto \color{blue}{-1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 90.2%

         \[\leadsto \color{blue}{-\cos re \cdot im} \]

      2. *-commutative 90.2%

         \[\leadsto -\color{blue}{im \cdot \cos re} \]

      3. distribute-lft-neg-in 90.2%

         \[\leadsto \color{blue}{\left(-im\right) \cdot \cos re} \]

   6. Simplified 90.2%

      \[\leadsto \color{blue}{\left(-im\right) \cdot \cos re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.32 \cdot 10^{+47} \lor \neg \left(im \leq 9 \cdot 10^{+46}\right):\\ \;\;\;\;{im}^{3} \cdot -0.16666666666666666 - im\\ \mathbf{else}:\\ \;\;\;\;im \cdot \left(-\cos re\right)\\ \end{array} \]

Alternative 9: 34.6% accurate, 27.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(re \cdot re\right) \cdot -6.75\\ \mathbf{if}\;im \leq -5.6 \cdot 10^{+172}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;im \leq -4.1 \cdot 10^{+18}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \mathbf{elif}\;im \leq 500:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (let* ((t_0 (* (* re re) -6.75)))
   (if (<= im -5.6e+172)
     t_0
     (if (<= im -4.1e+18) (* (* re re) 0.75) (if (<= im 500.0) (- im) t_0)))))

C

double code(double re, double im) {
	double t_0 = (re * re) * -6.75;
	double tmp;
	if (im <= -5.6e+172) {
		tmp = t_0;
	} else if (im <= -4.1e+18) {
		tmp = (re * re) * 0.75;
	} else if (im <= 500.0) {
		tmp = -im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (re * re) * (-6.75d0)
    if (im <= (-5.6d+172)) then
        tmp = t_0
    else if (im <= (-4.1d+18)) then
        tmp = (re * re) * 0.75d0
    else if (im <= 500.0d0) then
        tmp = -im
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double t_0 = (re * re) * -6.75;
	double tmp;
	if (im <= -5.6e+172) {
		tmp = t_0;
	} else if (im <= -4.1e+18) {
		tmp = (re * re) * 0.75;
	} else if (im <= 500.0) {
		tmp = -im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(re, im):
	t_0 = (re * re) * -6.75
	tmp = 0
	if im <= -5.6e+172:
		tmp = t_0
	elif im <= -4.1e+18:
		tmp = (re * re) * 0.75
	elif im <= 500.0:
		tmp = -im
	else:
		tmp = t_0
	return tmp

Julia

function code(re, im)
	t_0 = Float64(Float64(re * re) * -6.75)
	tmp = 0.0
	if (im <= -5.6e+172)
		tmp = t_0;
	elseif (im <= -4.1e+18)
		tmp = Float64(Float64(re * re) * 0.75);
	elseif (im <= 500.0)
		tmp = Float64(-im);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	t_0 = (re * re) * -6.75;
	tmp = 0.0;
	if (im <= -5.6e+172)
		tmp = t_0;
	elseif (im <= -4.1e+18)
		tmp = (re * re) * 0.75;
	elseif (im <= 500.0)
		tmp = -im;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := Block[{t$95$0 = N[(N[(re * re), $MachinePrecision] * -6.75), $MachinePrecision]}, If[LessEqual[im, -5.6e+172], t$95$0, If[LessEqual[im, -4.1e+18], N[(N[(re * re), $MachinePrecision] * 0.75), $MachinePrecision], If[LessEqual[im, 500.0], (-im), t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if im < -5.5999999999999999e172 or 500 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right) + 0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.25 \cdot \color{blue}{\left({re}^{2} \cdot \left(e^{-im} - e^{im}\right)\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      2. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{-im} - e^{im}\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      3. distribute-rgt-out 74.8%

         \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      4. +-commutative 74.8%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. *-commutative 74.8%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{{re}^{2} \cdot -0.25}\right) \]

      6. unpow2 74.8%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{\left(re \cdot re\right)} \cdot -0.25\right) \]

      7. associate-*l* 74.8%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{re \cdot \left(re \cdot -0.25\right)}\right) \]

   6. Simplified 74.8%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)} \]

   7. Taylor expanded in re around inf 23.3%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 23.3%

         \[\leadsto \color{blue}{\left(-0.25 \cdot \left(e^{-im} - e^{im}\right)\right) \cdot {re}^{2}} \]

      2. *-commutative 23.3%

         \[\leadsto \color{blue}{{re}^{2} \cdot \left(-0.25 \cdot \left(e^{-im} - e^{im}\right)\right)} \]

      3. unpow2 23.3%

         \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot \left(-0.25 \cdot \left(e^{-im} - e^{im}\right)\right) \]

   9. Simplified 23.3%

      \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(-0.25 \cdot \left(e^{-im} - e^{im}\right)\right)} \]

   11. Applied egg-rr 21.1%

       \[\leadsto \left(re \cdot re\right) \cdot \left(-0.25 \cdot \color{blue}{27}\right) \]

   if -5.5999999999999999e172 < im < -4.1e18

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right) + 0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.25 \cdot \color{blue}{\left({re}^{2} \cdot \left(e^{-im} - e^{im}\right)\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      2. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{-im} - e^{im}\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      3. distribute-rgt-out 56.8%

         \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      4. +-commutative 56.8%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. *-commutative 56.8%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{{re}^{2} \cdot -0.25}\right) \]

      6. unpow2 56.8%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{\left(re \cdot re\right)} \cdot -0.25\right) \]

      7. associate-*l* 56.8%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{re \cdot \left(re \cdot -0.25\right)}\right) \]

   6. Simplified 56.8%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)} \]

   8. Applied egg-rr 25.6%

      \[\leadsto \color{blue}{-3} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

   9. Taylor expanded in re around inf 26.2%

      \[\leadsto \color{blue}{0.75 \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. *-commutative 26.2%

          \[\leadsto \color{blue}{{re}^{2} \cdot 0.75} \]

       2. unpow2 26.2%

          \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot 0.75 \]

   11. Simplified 26.2%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot 0.75} \]

   if -4.1e18 < im < 500

   1. Initial program 8.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 8.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 8.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in im around 0 97.7%

      \[\leadsto \color{blue}{-1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 97.7%

         \[\leadsto \color{blue}{-\cos re \cdot im} \]

      2. *-commutative 97.7%

         \[\leadsto -\color{blue}{im \cdot \cos re} \]

      3. distribute-lft-neg-in 97.7%

         \[\leadsto \color{blue}{\left(-im\right) \cdot \cos re} \]

   6. Simplified 97.7%

      \[\leadsto \color{blue}{\left(-im\right) \cdot \cos re} \]

   7. Taylor expanded in re around 0 54.0%

      \[\leadsto \color{blue}{-1 \cdot im} \]
   8. Step-by-step derivation

      1. mul-1-neg 54.0%

         \[\leadsto \color{blue}{-im} \]

   9. Simplified 54.0%

      \[\leadsto \color{blue}{-im} \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -5.6 \cdot 10^{+172}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \mathbf{elif}\;im \leq -4.1 \cdot 10^{+18}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \mathbf{elif}\;im \leq 500:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \end{array} \]

Alternative 10: 35.5% accurate, 27.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -1.45 \cdot 10^{+173}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \mathbf{elif}\;im \leq -4.6 \cdot 10^{+18}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \mathbf{elif}\;im \leq 580:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -1.45e+173)
   (* (* re re) (* im 0.5))
   (if (<= im -4.6e+18)
     (* (* re re) 0.75)
     (if (<= im 580.0) (- im) (* (* re re) -6.75)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -1.45e+173) {
		tmp = (re * re) * (im * 0.5);
	} else if (im <= -4.6e+18) {
		tmp = (re * re) * 0.75;
	} else if (im <= 580.0) {
		tmp = -im;
	} else {
		tmp = (re * re) * -6.75;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-1.45d+173)) then
        tmp = (re * re) * (im * 0.5d0)
    else if (im <= (-4.6d+18)) then
        tmp = (re * re) * 0.75d0
    else if (im <= 580.0d0) then
        tmp = -im
    else
        tmp = (re * re) * (-6.75d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -1.45e+173) {
		tmp = (re * re) * (im * 0.5);
	} else if (im <= -4.6e+18) {
		tmp = (re * re) * 0.75;
	} else if (im <= 580.0) {
		tmp = -im;
	} else {
		tmp = (re * re) * -6.75;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -1.45e+173:
		tmp = (re * re) * (im * 0.5)
	elif im <= -4.6e+18:
		tmp = (re * re) * 0.75
	elif im <= 580.0:
		tmp = -im
	else:
		tmp = (re * re) * -6.75
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -1.45e+173)
		tmp = Float64(Float64(re * re) * Float64(im * 0.5));
	elseif (im <= -4.6e+18)
		tmp = Float64(Float64(re * re) * 0.75);
	elseif (im <= 580.0)
		tmp = Float64(-im);
	else
		tmp = Float64(Float64(re * re) * -6.75);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -1.45e+173)
		tmp = (re * re) * (im * 0.5);
	elseif (im <= -4.6e+18)
		tmp = (re * re) * 0.75;
	elseif (im <= 580.0)
		tmp = -im;
	else
		tmp = (re * re) * -6.75;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -1.45e+173], N[(N[(re * re), $MachinePrecision] * N[(im * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[im, -4.6e+18], N[(N[(re * re), $MachinePrecision] * 0.75), $MachinePrecision], If[LessEqual[im, 580.0], (-im), N[(N[(re * re), $MachinePrecision] * -6.75), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < -1.45000000000000003e173

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right) + 0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.25 \cdot \color{blue}{\left({re}^{2} \cdot \left(e^{-im} - e^{im}\right)\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      2. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{-im} - e^{im}\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      3. distribute-rgt-out 82.8%

         \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      4. +-commutative 82.8%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. *-commutative 82.8%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{{re}^{2} \cdot -0.25}\right) \]

      6. unpow2 82.8%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{\left(re \cdot re\right)} \cdot -0.25\right) \]

      7. associate-*l* 82.8%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{re \cdot \left(re \cdot -0.25\right)}\right) \]

   6. Simplified 82.8%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)} \]

   7. Taylor expanded in re around inf 41.4%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 41.4%

         \[\leadsto \color{blue}{\left(-0.25 \cdot \left(e^{-im} - e^{im}\right)\right) \cdot {re}^{2}} \]

      2. *-commutative 41.4%

         \[\leadsto \color{blue}{{re}^{2} \cdot \left(-0.25 \cdot \left(e^{-im} - e^{im}\right)\right)} \]

      3. unpow2 41.4%

         \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot \left(-0.25 \cdot \left(e^{-im} - e^{im}\right)\right) \]

   9. Simplified 41.4%

      \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(-0.25 \cdot \left(e^{-im} - e^{im}\right)\right)} \]

   10. Taylor expanded in im around 0 41.9%

       \[\leadsto \color{blue}{0.5 \cdot \left({re}^{2} \cdot im\right)} \]
   11. Step-by-step derivation

       1. *-commutative 41.9%

          \[\leadsto \color{blue}{\left({re}^{2} \cdot im\right) \cdot 0.5} \]

       2. associate-*l* 41.9%

          \[\leadsto \color{blue}{{re}^{2} \cdot \left(im \cdot 0.5\right)} \]

       3. unpow2 41.9%

          \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot \left(im \cdot 0.5\right) \]

   12. Simplified 41.9%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)} \]

   if -1.45000000000000003e173 < im < -4.6e18

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right) + 0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.25 \cdot \color{blue}{\left({re}^{2} \cdot \left(e^{-im} - e^{im}\right)\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      2. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{-im} - e^{im}\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      3. distribute-rgt-out 56.8%

         \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      4. +-commutative 56.8%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. *-commutative 56.8%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{{re}^{2} \cdot -0.25}\right) \]

      6. unpow2 56.8%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{\left(re \cdot re\right)} \cdot -0.25\right) \]

      7. associate-*l* 56.8%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{re \cdot \left(re \cdot -0.25\right)}\right) \]

   6. Simplified 56.8%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)} \]

   8. Applied egg-rr 25.6%

      \[\leadsto \color{blue}{-3} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

   9. Taylor expanded in re around inf 26.2%

      \[\leadsto \color{blue}{0.75 \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. *-commutative 26.2%

          \[\leadsto \color{blue}{{re}^{2} \cdot 0.75} \]

       2. unpow2 26.2%

          \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot 0.75 \]

   11. Simplified 26.2%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot 0.75} \]

   if -4.6e18 < im < 580

   1. Initial program 8.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 8.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 8.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in im around 0 97.7%

      \[\leadsto \color{blue}{-1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 97.7%

         \[\leadsto \color{blue}{-\cos re \cdot im} \]

      2. *-commutative 97.7%

         \[\leadsto -\color{blue}{im \cdot \cos re} \]

      3. distribute-lft-neg-in 97.7%

         \[\leadsto \color{blue}{\left(-im\right) \cdot \cos re} \]

   6. Simplified 97.7%

      \[\leadsto \color{blue}{\left(-im\right) \cdot \cos re} \]

   7. Taylor expanded in re around 0 54.0%

      \[\leadsto \color{blue}{-1 \cdot im} \]
   8. Step-by-step derivation

      1. mul-1-neg 54.0%

         \[\leadsto \color{blue}{-im} \]

   9. Simplified 54.0%

      \[\leadsto \color{blue}{-im} \]

   if 580 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right) + 0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.25 \cdot \color{blue}{\left({re}^{2} \cdot \left(e^{-im} - e^{im}\right)\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      2. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{-im} - e^{im}\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      3. distribute-rgt-out 71.6%

         \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      4. +-commutative 71.6%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. *-commutative 71.6%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{{re}^{2} \cdot -0.25}\right) \]

      6. unpow2 71.6%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{\left(re \cdot re\right)} \cdot -0.25\right) \]

      7. associate-*l* 71.6%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{re \cdot \left(re \cdot -0.25\right)}\right) \]

   6. Simplified 71.6%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)} \]

   7. Taylor expanded in re around inf 16.2%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 16.2%

         \[\leadsto \color{blue}{\left(-0.25 \cdot \left(e^{-im} - e^{im}\right)\right) \cdot {re}^{2}} \]

      2. *-commutative 16.2%

         \[\leadsto \color{blue}{{re}^{2} \cdot \left(-0.25 \cdot \left(e^{-im} - e^{im}\right)\right)} \]

      3. unpow2 16.2%

         \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot \left(-0.25 \cdot \left(e^{-im} - e^{im}\right)\right) \]

   9. Simplified 16.2%

      \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(-0.25 \cdot \left(e^{-im} - e^{im}\right)\right)} \]

   11. Applied egg-rr 19.2%

       \[\leadsto \left(re \cdot re\right) \cdot \left(-0.25 \cdot \color{blue}{27}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 38.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -1.45 \cdot 10^{+173}:\\ \;\;\;\;\left(re \cdot re\right) \cdot \left(im \cdot 0.5\right)\\ \mathbf{elif}\;im \leq -4.6 \cdot 10^{+18}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \mathbf{elif}\;im \leq 580:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \end{array} \]

Alternative 11: 35.9% accurate, 27.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;im \leq -2.3 \cdot 10^{+171}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot re\right)\right) - im\\ \mathbf{elif}\;im \leq -4.1 \cdot 10^{+18}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \mathbf{elif}\;im \leq 520:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (<= im -2.3e+171)
   (- (* 0.5 (* re (* im re))) im)
   (if (<= im -4.1e+18)
     (* (* re re) 0.75)
     (if (<= im 520.0) (- im) (* (* re re) -6.75)))))

C

double code(double re, double im) {
	double tmp;
	if (im <= -2.3e+171) {
		tmp = (0.5 * (re * (im * re))) - im;
	} else if (im <= -4.1e+18) {
		tmp = (re * re) * 0.75;
	} else if (im <= 520.0) {
		tmp = -im;
	} else {
		tmp = (re * re) * -6.75;
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (im <= (-2.3d+171)) then
        tmp = (0.5d0 * (re * (im * re))) - im
    else if (im <= (-4.1d+18)) then
        tmp = (re * re) * 0.75d0
    else if (im <= 520.0d0) then
        tmp = -im
    else
        tmp = (re * re) * (-6.75d0)
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (im <= -2.3e+171) {
		tmp = (0.5 * (re * (im * re))) - im;
	} else if (im <= -4.1e+18) {
		tmp = (re * re) * 0.75;
	} else if (im <= 520.0) {
		tmp = -im;
	} else {
		tmp = (re * re) * -6.75;
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if im <= -2.3e+171:
		tmp = (0.5 * (re * (im * re))) - im
	elif im <= -4.1e+18:
		tmp = (re * re) * 0.75
	elif im <= 520.0:
		tmp = -im
	else:
		tmp = (re * re) * -6.75
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (im <= -2.3e+171)
		tmp = Float64(Float64(0.5 * Float64(re * Float64(im * re))) - im);
	elseif (im <= -4.1e+18)
		tmp = Float64(Float64(re * re) * 0.75);
	elseif (im <= 520.0)
		tmp = Float64(-im);
	else
		tmp = Float64(Float64(re * re) * -6.75);
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (im <= -2.3e+171)
		tmp = (0.5 * (re * (im * re))) - im;
	elseif (im <= -4.1e+18)
		tmp = (re * re) * 0.75;
	elseif (im <= 520.0)
		tmp = -im;
	else
		tmp = (re * re) * -6.75;
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[LessEqual[im, -2.3e+171], N[(N[(0.5 * N[(re * N[(im * re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - im), $MachinePrecision], If[LessEqual[im, -4.1e+18], N[(N[(re * re), $MachinePrecision] * 0.75), $MachinePrecision], If[LessEqual[im, 520.0], (-im), N[(N[(re * re), $MachinePrecision] * -6.75), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if im < -2.30000000000000017e171

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in im around 0 7.0%

      \[\leadsto \color{blue}{-1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 7.0%

         \[\leadsto \color{blue}{-\cos re \cdot im} \]

      2. *-commutative 7.0%

         \[\leadsto -\color{blue}{im \cdot \cos re} \]

      3. distribute-lft-neg-in 7.0%

         \[\leadsto \color{blue}{\left(-im\right) \cdot \cos re} \]

   6. Simplified 7.0%

      \[\leadsto \color{blue}{\left(-im\right) \cdot \cos re} \]

   7. Taylor expanded in re around 0 43.1%

      \[\leadsto \color{blue}{-1 \cdot im + 0.5 \cdot \left({re}^{2} \cdot im\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 43.1%

         \[\leadsto \color{blue}{\left(-im\right)} + 0.5 \cdot \left({re}^{2} \cdot im\right) \]

      2. +-commutative 43.1%

         \[\leadsto \color{blue}{0.5 \cdot \left({re}^{2} \cdot im\right) + \left(-im\right)} \]

      3. unsub-neg 43.1%

         \[\leadsto \color{blue}{0.5 \cdot \left({re}^{2} \cdot im\right) - im} \]

      4. unpow2 43.1%

         \[\leadsto 0.5 \cdot \left(\color{blue}{\left(re \cdot re\right)} \cdot im\right) - im \]

      5. associate-*l* 43.1%

         \[\leadsto 0.5 \cdot \color{blue}{\left(re \cdot \left(re \cdot im\right)\right)} - im \]

   9. Simplified 43.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(re \cdot \left(re \cdot im\right)\right) - im} \]

   if -2.30000000000000017e171 < im < -4.1e18

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right) + 0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.25 \cdot \color{blue}{\left({re}^{2} \cdot \left(e^{-im} - e^{im}\right)\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      2. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{-im} - e^{im}\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      3. distribute-rgt-out 55.6%

         \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      4. +-commutative 55.6%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. *-commutative 55.6%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{{re}^{2} \cdot -0.25}\right) \]

      6. unpow2 55.6%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{\left(re \cdot re\right)} \cdot -0.25\right) \]

      7. associate-*l* 55.6%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{re \cdot \left(re \cdot -0.25\right)}\right) \]

   6. Simplified 55.6%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)} \]

   8. Applied egg-rr 26.3%

      \[\leadsto \color{blue}{-3} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

   9. Taylor expanded in re around inf 26.9%

      \[\leadsto \color{blue}{0.75 \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. *-commutative 26.9%

          \[\leadsto \color{blue}{{re}^{2} \cdot 0.75} \]

       2. unpow2 26.9%

          \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot 0.75 \]

   11. Simplified 26.9%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot 0.75} \]

   if -4.1e18 < im < 520

   1. Initial program 8.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 8.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 8.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in im around 0 97.7%

      \[\leadsto \color{blue}{-1 \cdot \left(\cos re \cdot im\right)} \]
   5. Step-by-step derivation

      1. mul-1-neg 97.7%

         \[\leadsto \color{blue}{-\cos re \cdot im} \]

      2. *-commutative 97.7%

         \[\leadsto -\color{blue}{im \cdot \cos re} \]

      3. distribute-lft-neg-in 97.7%

         \[\leadsto \color{blue}{\left(-im\right) \cdot \cos re} \]

   6. Simplified 97.7%

      \[\leadsto \color{blue}{\left(-im\right) \cdot \cos re} \]

   7. Taylor expanded in re around 0 54.0%

      \[\leadsto \color{blue}{-1 \cdot im} \]
   8. Step-by-step derivation

      1. mul-1-neg 54.0%

         \[\leadsto \color{blue}{-im} \]

   9. Simplified 54.0%

      \[\leadsto \color{blue}{-im} \]

   if 520 < im

   1. Initial program 100.0%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 100.0%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right) + 0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.25 \cdot \color{blue}{\left({re}^{2} \cdot \left(e^{-im} - e^{im}\right)\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      2. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{-im} - e^{im}\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      3. distribute-rgt-out 71.6%

         \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      4. +-commutative 71.6%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. *-commutative 71.6%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{{re}^{2} \cdot -0.25}\right) \]

      6. unpow2 71.6%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{\left(re \cdot re\right)} \cdot -0.25\right) \]

      7. associate-*l* 71.6%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{re \cdot \left(re \cdot -0.25\right)}\right) \]

   6. Simplified 71.6%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)} \]

   7. Taylor expanded in re around inf 16.2%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 16.2%

         \[\leadsto \color{blue}{\left(-0.25 \cdot \left(e^{-im} - e^{im}\right)\right) \cdot {re}^{2}} \]

      2. *-commutative 16.2%

         \[\leadsto \color{blue}{{re}^{2} \cdot \left(-0.25 \cdot \left(e^{-im} - e^{im}\right)\right)} \]

      3. unpow2 16.2%

         \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot \left(-0.25 \cdot \left(e^{-im} - e^{im}\right)\right) \]

   9. Simplified 16.2%

      \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(-0.25 \cdot \left(e^{-im} - e^{im}\right)\right)} \]

   11. Applied egg-rr 19.2%

       \[\leadsto \left(re \cdot re\right) \cdot \left(-0.25 \cdot \color{blue}{27}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;im \leq -2.3 \cdot 10^{+171}:\\ \;\;\;\;0.5 \cdot \left(re \cdot \left(im \cdot re\right)\right) - im\\ \mathbf{elif}\;im \leq -4.1 \cdot 10^{+18}:\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \mathbf{elif}\;im \leq 520:\\ \;\;\;\;-im\\ \mathbf{else}:\\ \;\;\;\;\left(re \cdot re\right) \cdot -6.75\\ \end{array} \]

Alternative 12: 34.2% accurate, 33.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;re \leq -5.2 \cdot 10^{+160} \lor \neg \left(re \leq 1.15 \cdot 10^{+176}\right):\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (or (<= re -5.2e+160) (not (<= re 1.15e+176)))
   (* (* re re) 0.75)
   (* 0.5 (* im -2.0))))

C

double code(double re, double im) {
	double tmp;
	if ((re <= -5.2e+160) || !(re <= 1.15e+176)) {
		tmp = (re * re) * 0.75;
	} else {
		tmp = 0.5 * (im * -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if ((re <= (-5.2d+160)) .or. (.not. (re <= 1.15d+176))) then
        tmp = (re * re) * 0.75d0
    else
        tmp = 0.5d0 * (im * (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if ((re <= -5.2e+160) || !(re <= 1.15e+176)) {
		tmp = (re * re) * 0.75;
	} else {
		tmp = 0.5 * (im * -2.0);
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if (re <= -5.2e+160) or not (re <= 1.15e+176):
		tmp = (re * re) * 0.75
	else:
		tmp = 0.5 * (im * -2.0)
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if ((re <= -5.2e+160) || !(re <= 1.15e+176))
		tmp = Float64(Float64(re * re) * 0.75);
	else
		tmp = Float64(0.5 * Float64(im * -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if ((re <= -5.2e+160) || ~((re <= 1.15e+176)))
		tmp = (re * re) * 0.75;
	else
		tmp = 0.5 * (im * -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Or[LessEqual[re, -5.2e+160], N[Not[LessEqual[re, 1.15e+176]], $MachinePrecision]], N[(N[(re * re), $MachinePrecision] * 0.75), $MachinePrecision], N[(0.5 * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if re < -5.2000000000000001e160 or 1.14999999999999998e176 < re

   1. Initial program 63.3%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 63.3%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 63.3%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in re around 0 0.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right) + 0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 0.0%

         \[\leadsto -0.25 \cdot \color{blue}{\left({re}^{2} \cdot \left(e^{-im} - e^{im}\right)\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      2. associate-*r* 0.0%

         \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{-im} - e^{im}\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

      3. distribute-rgt-out 23.7%

         \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

      4. +-commutative 23.7%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

      5. *-commutative 23.7%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{{re}^{2} \cdot -0.25}\right) \]

      6. unpow2 23.7%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{\left(re \cdot re\right)} \cdot -0.25\right) \]

      7. associate-*l* 23.7%

         \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{re \cdot \left(re \cdot -0.25\right)}\right) \]

   6. Simplified 23.7%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)} \]

   8. Applied egg-rr 27.7%

      \[\leadsto \color{blue}{-3} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

   9. Taylor expanded in re around inf 27.7%

      \[\leadsto \color{blue}{0.75 \cdot {re}^{2}} \]
   10. Step-by-step derivation

       1. *-commutative 27.7%

          \[\leadsto \color{blue}{{re}^{2} \cdot 0.75} \]

       2. unpow2 27.7%

          \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot 0.75 \]

   11. Simplified 27.7%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot 0.75} \]

   if -5.2000000000000001e160 < re < 1.14999999999999998e176

   1. Initial program 56.8%

      \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
   2. Step-by-step derivation

      1. sub0-neg 56.8%

         \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

   3. Simplified 56.8%

      \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

   4. Taylor expanded in re around 0 48.2%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-im} - e^{im}\right)} \]

   5. Taylor expanded in im around 0 33.0%

      \[\leadsto 0.5 \cdot \color{blue}{\left(-2 \cdot im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \leq -5.2 \cdot 10^{+160} \lor \neg \left(re \leq 1.15 \cdot 10^{+176}\right):\\ \;\;\;\;\left(re \cdot re\right) \cdot 0.75\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(im \cdot -2\right)\\ \end{array} \]

Alternative 13: 30.3% accurate, 61.8× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(im \cdot -2\right) \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (* 0.5 (* im -2.0)))

C

double code(double re, double im) {
	return 0.5 * (im * -2.0);
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.5d0 * (im * (-2.0d0))
end function

Java

public static double code(double re, double im) {
	return 0.5 * (im * -2.0);
}

Python

def code(re, im):
	return 0.5 * (im * -2.0)

Julia

function code(re, im)
	return Float64(0.5 * Float64(im * -2.0))
end

MATLAB

function tmp = code(re, im)
	tmp = 0.5 * (im * -2.0);
end

Wolfram

code[re_, im_] := N[(0.5 * N[(im * -2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.3%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Step-by-step derivation

   1. sub0-neg 58.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

3. Simplified 58.3%

   \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

4. Taylor expanded in re around 0 46.3%

   \[\leadsto \color{blue}{0.5 \cdot \left(e^{-im} - e^{im}\right)} \]

5. Taylor expanded in im around 0 27.2%

   \[\leadsto 0.5 \cdot \color{blue}{\left(-2 \cdot im\right)} \]

6. Final simplification 27.2%

   \[\leadsto 0.5 \cdot \left(im \cdot -2\right) \]

Alternative 14: 30.3% accurate, 154.5× speedup

Math

\[\begin{array}{l} \\ -im \end{array} \]

FPCore

(FPCore (re im) :precision binary64 (- im))

C

double code(double re, double im) {
	return -im;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -im
end function

Java

public static double code(double re, double im) {
	return -im;
}

Python

def code(re, im):
	return -im

Julia

function code(re, im)
	return Float64(-im)
end

MATLAB

function tmp = code(re, im)
	tmp = -im;
end

Wolfram

code[re_, im_] := (-im)

Derivation

1. Initial program 58.3%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Step-by-step derivation

   1. sub0-neg 58.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

3. Simplified 58.3%

   \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

4. Taylor expanded in im around 0 47.3%

   \[\leadsto \color{blue}{-1 \cdot \left(\cos re \cdot im\right)} \]
5. Step-by-step derivation

   1. mul-1-neg 47.3%

      \[\leadsto \color{blue}{-\cos re \cdot im} \]

   2. *-commutative 47.3%

      \[\leadsto -\color{blue}{im \cdot \cos re} \]

   3. distribute-lft-neg-in 47.3%

      \[\leadsto \color{blue}{\left(-im\right) \cdot \cos re} \]

6. Simplified 47.3%

   \[\leadsto \color{blue}{\left(-im\right) \cdot \cos re} \]

7. Taylor expanded in re around 0 26.9%

   \[\leadsto \color{blue}{-1 \cdot im} \]
8. Step-by-step derivation

   1. mul-1-neg 26.9%

      \[\leadsto \color{blue}{-im} \]

9. Simplified 26.9%

   \[\leadsto \color{blue}{-im} \]

10. Final simplification 26.9%

    \[\leadsto -im \]

Alternative 15: 2.9% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ -1.5 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 -1.5)

C

double code(double re, double im) {
	return -1.5;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = -1.5d0
end function

Java

public static double code(double re, double im) {
	return -1.5;
}

Python

def code(re, im):
	return -1.5

Julia

function code(re, im)
	return -1.5
end

MATLAB

function tmp = code(re, im)
	tmp = -1.5;
end

Wolfram

code[re_, im_] := -1.5

Derivation

1. Initial program 58.3%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Step-by-step derivation

   1. sub0-neg 58.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

3. Simplified 58.3%

   \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

4. Taylor expanded in re around 0 2.0%

   \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right) + 0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
5. Step-by-step derivation

   1. *-commutative 2.0%

      \[\leadsto -0.25 \cdot \color{blue}{\left({re}^{2} \cdot \left(e^{-im} - e^{im}\right)\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

   2. associate-*r* 2.0%

      \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{-im} - e^{im}\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

   3. distribute-rgt-out 41.0%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

   4. +-commutative 41.0%

      \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

   5. *-commutative 41.0%

      \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{{re}^{2} \cdot -0.25}\right) \]

   6. unpow2 41.0%

      \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{\left(re \cdot re\right)} \cdot -0.25\right) \]

   7. associate-*l* 41.0%

      \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{re \cdot \left(re \cdot -0.25\right)}\right) \]

6. Simplified 41.0%

   \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)} \]

8. Applied egg-rr 8.5%

   \[\leadsto \color{blue}{-3} \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right) \]

9. Taylor expanded in re around 0 2.9%

   \[\leadsto \color{blue}{-1.5} \]

10. Final simplification 2.9%

    \[\leadsto -1.5 \]

Alternative 16: 3.5% accurate, 309.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (re im) :precision binary64 0.0)

C

double code(double re, double im) {
	return 0.0;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    code = 0.0d0
end function

Java

public static double code(double re, double im) {
	return 0.0;
}

Python

def code(re, im):
	return 0.0

Julia

function code(re, im)
	return 0.0
end

MATLAB

function tmp = code(re, im)
	tmp = 0.0;
end

Wolfram

code[re_, im_] := 0.0

Derivation

1. Initial program 58.3%

   \[\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right) \]
2. Step-by-step derivation

   1. sub0-neg 58.3%

      \[\leadsto \left(0.5 \cdot \cos re\right) \cdot \left(e^{\color{blue}{-im}} - e^{im}\right) \]

3. Simplified 58.3%

   \[\leadsto \color{blue}{\left(0.5 \cdot \cos re\right) \cdot \left(e^{-im} - e^{im}\right)} \]

4. Taylor expanded in re around 0 2.0%

   \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right) + 0.5 \cdot \left(e^{-im} - e^{im}\right)} \]
5. Step-by-step derivation

   1. *-commutative 2.0%

      \[\leadsto -0.25 \cdot \color{blue}{\left({re}^{2} \cdot \left(e^{-im} - e^{im}\right)\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

   2. associate-*r* 2.0%

      \[\leadsto \color{blue}{\left(-0.25 \cdot {re}^{2}\right) \cdot \left(e^{-im} - e^{im}\right)} + 0.5 \cdot \left(e^{-im} - e^{im}\right) \]

   3. distribute-rgt-out 41.0%

      \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(-0.25 \cdot {re}^{2} + 0.5\right)} \]

   4. +-commutative 41.0%

      \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \color{blue}{\left(0.5 + -0.25 \cdot {re}^{2}\right)} \]

   5. *-commutative 41.0%

      \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{{re}^{2} \cdot -0.25}\right) \]

   6. unpow2 41.0%

      \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{\left(re \cdot re\right)} \cdot -0.25\right) \]

   7. associate-*l* 41.0%

      \[\leadsto \left(e^{-im} - e^{im}\right) \cdot \left(0.5 + \color{blue}{re \cdot \left(re \cdot -0.25\right)}\right) \]

6. Simplified 41.0%

   \[\leadsto \color{blue}{\left(e^{-im} - e^{im}\right) \cdot \left(0.5 + re \cdot \left(re \cdot -0.25\right)\right)} \]

7. Taylor expanded in re around inf 13.7%

   \[\leadsto \color{blue}{-0.25 \cdot \left(\left(e^{-im} - e^{im}\right) \cdot {re}^{2}\right)} \]
8. Step-by-step derivation

   1. associate-*r* 13.7%

      \[\leadsto \color{blue}{\left(-0.25 \cdot \left(e^{-im} - e^{im}\right)\right) \cdot {re}^{2}} \]

   2. *-commutative 13.7%

      \[\leadsto \color{blue}{{re}^{2} \cdot \left(-0.25 \cdot \left(e^{-im} - e^{im}\right)\right)} \]

   3. unpow2 13.7%

      \[\leadsto \color{blue}{\left(re \cdot re\right)} \cdot \left(-0.25 \cdot \left(e^{-im} - e^{im}\right)\right) \]

9. Simplified 13.7%

   \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(-0.25 \cdot \left(e^{-im} - e^{im}\right)\right)} \]
10. Step-by-step derivation

    1. expm1-log1p-u 6.6%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(re \cdot re\right) \cdot \left(-0.25 \cdot \left(e^{-im} - e^{im}\right)\right)\right)\right)} \]

    2. expm1-udef 6.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(re \cdot re\right) \cdot \left(-0.25 \cdot \left(e^{-im} - e^{im}\right)\right)\right)} - 1} \]

    3. associate-*l* 7.2%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{re \cdot \left(re \cdot \left(-0.25 \cdot \left(e^{-im} - e^{im}\right)\right)\right)}\right)} - 1 \]

    4. add-sqr-sqrt 1.3%

       \[\leadsto e^{\mathsf{log1p}\left(re \cdot \left(re \cdot \left(-0.25 \cdot \left(e^{\color{blue}{\sqrt{-im} \cdot \sqrt{-im}}} - e^{im}\right)\right)\right)\right)} - 1 \]

    5. sqrt-unprod 2.5%

       \[\leadsto e^{\mathsf{log1p}\left(re \cdot \left(re \cdot \left(-0.25 \cdot \left(e^{\color{blue}{\sqrt{\left(-im\right) \cdot \left(-im\right)}}} - e^{im}\right)\right)\right)\right)} - 1 \]

    6. sqr-neg 2.5%

       \[\leadsto e^{\mathsf{log1p}\left(re \cdot \left(re \cdot \left(-0.25 \cdot \left(e^{\sqrt{\color{blue}{im \cdot im}}} - e^{im}\right)\right)\right)\right)} - 1 \]

    7. sqrt-unprod 1.2%

       \[\leadsto e^{\mathsf{log1p}\left(re \cdot \left(re \cdot \left(-0.25 \cdot \left(e^{\color{blue}{\sqrt{im} \cdot \sqrt{im}}} - e^{im}\right)\right)\right)\right)} - 1 \]

    8. add-sqr-sqrt 2.9%

       \[\leadsto e^{\mathsf{log1p}\left(re \cdot \left(re \cdot \left(-0.25 \cdot \left(e^{\color{blue}{im}} - e^{im}\right)\right)\right)\right)} - 1 \]

11. Applied egg-rr 2.9%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(re \cdot \left(re \cdot \left(-0.25 \cdot \left(e^{im} - e^{im}\right)\right)\right)\right)} - 1} \]
12. Step-by-step derivation

    1. expm1-def 2.9%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(re \cdot \left(re \cdot \left(-0.25 \cdot \left(e^{im} - e^{im}\right)\right)\right)\right)\right)} \]

    2. expm1-log1p 2.9%

       \[\leadsto \color{blue}{re \cdot \left(re \cdot \left(-0.25 \cdot \left(e^{im} - e^{im}\right)\right)\right)} \]

    3. associate-*r* 2.3%

       \[\leadsto \color{blue}{\left(re \cdot re\right) \cdot \left(-0.25 \cdot \left(e^{im} - e^{im}\right)\right)} \]

    4. unpow2 2.3%

       \[\leadsto \color{blue}{{re}^{2}} \cdot \left(-0.25 \cdot \left(e^{im} - e^{im}\right)\right) \]

    5. +-inverses 2.6%

       \[\leadsto {re}^{2} \cdot \left(-0.25 \cdot \color{blue}{0}\right) \]

    6. metadata-eval 2.6%

       \[\leadsto {re}^{2} \cdot \color{blue}{0} \]

    7. mul0-rgt 3.4%

       \[\leadsto \color{blue}{0} \]

13. Simplified 3.4%

    \[\leadsto \color{blue}{0} \]

14. Final simplification 3.4%

    \[\leadsto 0 \]

Developer target: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|im\right| < 1:\\ \;\;\;\;-\cos re \cdot \left(\left(im + \left(\left(0.16666666666666666 \cdot im\right) \cdot im\right) \cdot im\right) + \left(\left(\left(\left(0.008333333333333333 \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right) \cdot im\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \cos re\right) \cdot \left(e^{0 - im} - e^{im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (re im)
 :precision binary64
 (if (< (fabs im) 1.0)
   (-
    (*
     (cos re)
     (+
      (+ im (* (* (* 0.16666666666666666 im) im) im))
      (* (* (* (* (* 0.008333333333333333 im) im) im) im) im))))
   (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im)))))

C

double code(double re, double im) {
	double tmp;
	if (fabs(im) < 1.0) {
		tmp = -(cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	}
	return tmp;
}

Fortran

real(8) function code(re, im)
    real(8), intent (in) :: re
    real(8), intent (in) :: im
    real(8) :: tmp
    if (abs(im) < 1.0d0) then
        tmp = -(cos(re) * ((im + (((0.16666666666666666d0 * im) * im) * im)) + (((((0.008333333333333333d0 * im) * im) * im) * im) * im)))
    else
        tmp = (0.5d0 * cos(re)) * (exp((0.0d0 - im)) - exp(im))
    end if
    code = tmp
end function

Java

public static double code(double re, double im) {
	double tmp;
	if (Math.abs(im) < 1.0) {
		tmp = -(Math.cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	} else {
		tmp = (0.5 * Math.cos(re)) * (Math.exp((0.0 - im)) - Math.exp(im));
	}
	return tmp;
}

Python

def code(re, im):
	tmp = 0
	if math.fabs(im) < 1.0:
		tmp = -(math.cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)))
	else:
		tmp = (0.5 * math.cos(re)) * (math.exp((0.0 - im)) - math.exp(im))
	return tmp

Julia

function code(re, im)
	tmp = 0.0
	if (abs(im) < 1.0)
		tmp = Float64(-Float64(cos(re) * Float64(Float64(im + Float64(Float64(Float64(0.16666666666666666 * im) * im) * im)) + Float64(Float64(Float64(Float64(Float64(0.008333333333333333 * im) * im) * im) * im) * im))));
	else
		tmp = Float64(Float64(0.5 * cos(re)) * Float64(exp(Float64(0.0 - im)) - exp(im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(re, im)
	tmp = 0.0;
	if (abs(im) < 1.0)
		tmp = -(cos(re) * ((im + (((0.16666666666666666 * im) * im) * im)) + (((((0.008333333333333333 * im) * im) * im) * im) * im)));
	else
		tmp = (0.5 * cos(re)) * (exp((0.0 - im)) - exp(im));
	end
	tmp_2 = tmp;
end

Wolfram

code[re_, im_] := If[Less[N[Abs[im], $MachinePrecision], 1.0], (-N[(N[Cos[re], $MachinePrecision] * N[(N[(im + N[(N[(N[(0.16666666666666666 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(0.008333333333333333 * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision] * im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(0.5 * N[Cos[re], $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(0.0 - im), $MachinePrecision]], $MachinePrecision] - N[Exp[im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2023196 
(FPCore (re im)
  :name "math.sin on complex, imaginary part"
  :precision binary64

  :herbie-target
  (if (< (fabs im) 1.0) (- (* (cos re) (+ (+ im (* (* (* 0.16666666666666666 im) im) im)) (* (* (* (* (* 0.008333333333333333 im) im) im) im) im)))) (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))

  (* (* 0.5 (cos re)) (- (exp (- 0.0 im)) (exp im))))